Particle Physics. Dr M.A. Thomson. ν e. e + W + Z 0 e + e - W - Part II, Lent Term 2004 HANDOUT VI. Dr M.A. Thomson Lent 2004

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1 Particle Physics Dr M.A. Thomson e + W + Z 0 e + q q Part II, Lent Term 2004 HANDOUT VI

2 2 The Weak Interaction The WEAK interaction acconts for many ecays in particle physics e.g. μ! e νeνμ, fi! e νeνfi, ß +! μ νμ, n! pe νe,... Characterize by long lifetimes, small cross sections

3 Two types of WEAK interaction CHARGED CURRENT (CC) - W ± Bosons NEUTRAL CURRENT (NC) - Z 0 Boson 3 WEAK force meiate by MASSIVE VECTOR BOSONS: MW ο 80 GeV MZ0 ο 90 GeV e.g. the WEAK interactions of electrons an electron netrinos: Z 0 Z 0 W + BOSON SELF-INTERACTIONS e + e + Z 0 γ Z 0 W + γ W + γ W + W + Z 0 W + Z 0 W + γ W + also interactions with PHOTONS (W-bosons are charge)

4 4 Fermi Theory n G F p WEAK interaction taken to be a 4- fermion contact interaction i.e no propagator copling strength GF GF= GeV 2 Beta Decay in Fermi Theory G F n p Golen Rle : Phase Space : 2-boy vs. 3-boy TWO BODY FINAL STATE: N = =fi = = 2ßjM fi j 2 ρ(e) with ρ = N E E2 (2ß) 3 ΩE (neglecting final state masses). Only consier one of the particles since the other fixe by (E,~p) conservation THREE BODY FINAL STATE (e.g fi-ecay): 2 N = E2 ν (2ß) 3 Ω νe ν E 2 e (2ß) 3 Ω ee e now necessary to consier phase space of two of the particles - the thir is then given by (E,~p) conservation

5 In Nclear fi-ecay the energy release in the nclear transition, E0, is share between the electron, netrino an the recoil kinetic energy of the ncles: 5 E0 = E e + E ν + T recoil Since the ncles is mch more massive than the electron/netrino: E0 ß E e + E ν an the nclear recoil ensres momentm conservation. For a given electron energy E e : E ν = E0 N = N E0 E ν = E2 ν (2ß) 3 Ω ν E 2 e (2ß) 3 Ω ee e Assming isotropic ecay istribtions an integrating over Ω e Ω ν gives: N E0 = (4ß) 2 E2 ν (2ß) 3 = E2 ν E2 e 4ß 4 E e = (E 0 E e ) 2 E 2 e 4ß 4 E 2 e (2ß) 3 E e E e = 2ßjM fi j 2 (E 0 E e ) 2 E 2 e 4ß 4 = jm fi j 2 (E 0 E e ) 2 Ee 2 E e 2ß 3 E e

6 In FERMI theory take: jm fi j 2 = GF 2 fjm nclear j 2 6 where the nclear matrix element jm nclear j 2 acconts for the overlap of the nclear wave-fnctions, an f is the Colomb correction. Here assme jm nclear j 2 = (sper-allowe transition) an neglect f. ) = G 2 F E e 2ß 3 (E 0 E e ) 2 E 2 e = G F 2 2ß 3 Z E 0 = G F 2 0 2ß 3» E 5 0 = G F 2 E ß 3 (E0 E e ) 2 E 2 e E e 3 2E E5 0 5 SARGENT RULE: fi / E 5 e.g. see μ an fi ecay By stying lifetimes for nclear beta ecay (an applying necessary corrections, etermine strength of weak copling in FERMI theory: G fi = :36 ± 0: GeV 2

7 Beta-Decay Spectrm = G 2 F E e Plot of q E e E 2e 2ß 3 (E 0 E e ) 2 E 2 e verss (E0 E e ) (Krie plot) is linear 7 s E e E 2 e / (E0 E e ) For a non-zero netrino mass this is moifie to = G 2 F E e 2ß 3 (E 0 E e ) 2 E 2 e s m 2 ν E0 E 2 Resoltion ) Most recent reslts (999) Tritim fi ecay: m(νe) < 3 ev If netrinos have mass m( ) fi m(e). Why so small?

8 Netrino Scattering in Fermi Theory (inverse fi-ecay) 8 + n! p + e G F n p ff = 2ßjM fi j 2 N E ο 2ßGF 2 E2 e (2ß) 3 Ω ff ο GF 2 s where E e is the energy of the e in the centre-of-mass system an p s is the energy in the centre-of-mass system. In the Laboratory frame: s = 2E ν m n ( see Hanot I) ff( n) ο (E ν in MeV) 0 43 cm 2 Netrinos only interact WEAKLY ) have very small interaction cross-sections. Here WEAK implies that yo nee approximately 50 light-years of water to stop a MeV netrino. Commnication via netrino beams (á la Star Trek) non-trivial! However, as E ν! the cross-section ff(ν μ e ) can become large. Violates maximm vale allowe by conservation of probability at p s = 740 GeV (UNITARITY LIMIT) FERMI Theory breaks own at high energies

9 9 Weak Charge Crrent - W ± Boson Fermi theory breaks own at high energy. Tre interaction escribe by exchange of charge W-bosons (W± ) Fermi theory is the low energy (q 2 fi MW 2 ) EFFECTIVE theory of the WEAK interaction Beta-Decay: G F n p n p ν μ e Scattering: G F n p W q - M W At low energies q 2 << MW 2 : W-Boson propagator q2 MW 2! MW 2

10 Compare WEAK an QED interactions 0 WEAK INTERACTION QED g em W q - M W g=e γ q 2 g em CHARGED CURRENT WEAK INTERACTION MW 2 - i.e appears as the POINT-LIKE interaction of FERMI theory. For q 2 fi MW 2 propagator becomes Massive Propagator! short range MW = 80:4 ± 0: GeV Range ß MW ο0.002 fm Exchange Boson carries electro-magnetic charge FLAVOUR CHANGING! ONLY WEAK interaction changes flavor Parity Violating! ONLY WEAK interaction can violate parity conservation

11 COMPARE Fermi theory c.f. massive propagator gw G F For q 2 fi MW compare matrix elements: g 2 W M 2 W! GF GF is small becase MW is large. The precise relationship is: g 2 W 8M 2 W! G F p 2 The nmerical factors are partly of historical origin (e.g. see Perkins 4 th Eition, page 20). MW = 80:4 GeV an GF = : GeV 2 ) = 0:65 ) ff W = g2 W 4ß ß 30 The intrinsic strength of the WEAK interaction is greater than that of the electro-magnetic interaction. At low energies (low q 2 ) it appears weak e to the massive propagator. ff S ß 0:2, ff W ß 0:03, ff EM ß 0:0 sggestive of UNIFICATION of the forces

12 Netrino Scattering with a Massive W Boson Replace contact interaction by massive boson exchange iagram: 2 W q - M W ff q 2 = 32ß with jqj = 2Esin 2 where is the scattering angle. (e.g. similar to Hanot I p.36) Integrate to give: g 4 w (q 2 MW 2 ) 2 ff = G F 2 s ß ff = G F 2 MW 2 ß s fi MW 2 s fl MW 2 Total cross section now well behave at high energies.

13 Parity Violation in Beta Decay 3 Revision : Nclear Physics Uner Parity ^P: ~r! ~r ~p / r! ~p ~L = ~r ~p! ~L ~μ! ~μ ^P: Axial vectors e.g. ~L, ~μ o not change sign EXPERIMENT: Align 60 Co nclei at low temperatres with ~B fiel 60 Co! 60 Ni e νe Observe anglar istribtion of e relative to ~B. µ (E,p) P µ (E,-p) If parity conserve expect eqal nmbers of e parallel an anti-parallel to ~B. Experiment (C.S. W 956) showe clear asymmetry ) PARITY VIOLATION in WEAK interactions

14 Origin of Parity Violation 4 In the ltra-relativistic (massless) limit only LEFT-HANDED PARTICLES an RIGHT-HANDED ANTI-PARTICLES. participate in the WEAK (charge crrent) interaction. For massive fermions the weak interaction coples preferentially to LEFT-HANDED particles an RIGHT-HANDED anti-particles. Compare QED an WEAK interaction. e + γ µ + QED : e + e! μ + μ e + µ + e + e + µ + µ + WEAK INTERACTION : νee! e µ + e + From 6 possible SPIN assignments only 4 give non-zero contribtions to cross section From 6 possible SPIN assignments only gives a nonzero contribtion to cross section.! LR LR

15 EXAMPLE νee! e scattering in the centre-of-mass frame (s = E e + E ν = 2E e ) 5 ν θ e In massless limit - only one Helicity state contribtes: ff Ω = 2ßjM fij 2 E2 e = where s is centre-of-mass energy M fi = For q 2 fi M 2 W (2ß) 3 6ß 3 jm fij 2 s gw p2 2 q 2 M 2 W jm fi j = 2G F p 2 ( + cos ) cos 2 2 ff Ω = 8ß 2 G F 2 s( + cos ) 2 ff = G F 2 s 3ß

16 Parity Violation 6 The WEAK interaction treats LH an RH states ifferently an therefore can violate PARITY (i.e. the interaction Hamiltonian oes not commte with ^P) Parity ALWAYS conserve in STRONG/EM interactions P = Y i P i Y i>j ( ) L ij where P i is the intrinsic parity of the particle i an L ij is the orbital anglar momentm between particles i an j. Taking L ij = 0 J P P η π 0 π 0 π η π 0 BR = 32 % π 0 π 0 η J P P η π + π BR < 0. % Parity is sally violate in WEAK interactions J P P bt NOT ALWAYS! K + π + π K + π 0 s W + BR = 2 % π + J P P π + π - K + π + π - π K + π - BR = 6 % s W + π + π +

17 Weak Leptonic Decays 7 Mons are fnamental leptons (m μ ß 206m e ). Electro-magnetic ecay μ! e fl IS NOT observe; the EM interaction oes not change flavor. Only the WEAK charge crrent changes flavor. Mons ecay weakly : μ! e ν μ gw G F As m 2 μ fi M W 2 ) can se FERMI theory to calclate ecay with (analogos to fi ecay). FERMI theory gives ecay with proportional to m 5 μ (Sargent Rle): However more complicate phase space integration (previosly neglecte kinetic energy of recoiling ncles) gives μ = fi μ = G 2 F 92ß 3 m5 μ Mon mass an lifetime known with high precision. fi μ =(2:9703 ± 0:00004) 0 6 s Use mon ecay to fix strength of WEAK interaction GF GF =(:6632 ± 0:00002) 0 5 GeV 2 GF is one of the best etermine fnamental qantities in particle physics.

18 Universality of Weak Copling 8 Can compare GF measre from μ -ecay with that obtaine from fi-ecay gw n p From mon ecay measre: G μ =(:6632 ± 0:00002) 0 5 GeV 2 From fi-ecay measre: G fi =(:36 ± 0:003) 0 5 GeV 2 Taking ratio gives G fi G μ =0:974 ± 0:003 Concle that the strength of the weak interaction is almost the same for mons/electrons as for p/own qarks an we ll shortly come back to the origin of this ifference (cos c ) Can also test niversality of the WEAK interaction in fi -ecays, e.g. ν τ τ - gw

19 Ta Decays 9 The fi mass is relatively large, m fi = :777 GeV, an as m fi > fm μ ; m ß ; m ρ ; :::g there a nmber of possible ta ecay moes, e.g. τ - gw ν τ τ - Ta Branching Fractions: fi! e ν fi (7:8 ± 0: %) fi! μ νμ ν fi (7:3 ± 0: %) fi! harons (64:7 ± 0:2 %) gw ν τ - τ ν τ π- First compare ν τ gw τ - = μ!e = G F fi μ 92ß 3 m5 μ = fi fi Br(fi! e) fi!e = 0:78 2 gw GF 2 92ß 3 m5 fi If niversal strength of WEAK interaction expect fi fi fi μ = 0:78 m5 μ m 5 fi

20 m μ ; m fi ; fi μ are all precisely measre Using: m μ =05:658 MeV m fi =(777:0 ± 0:3) MeV fi μ =(2:9703 ± 0:00004) 0 6 s gives a PREDICTION of fi fi = 2:9 ± 0:0 0 3 s compare to MEASURED VALUE: fi fi = 2:9 ± 0:0 0 3 s Consistent with the preicte vale, i.e. 20 Same WEAK CC copling for μ an fi. Also compare ν τ ν τ τ - gw τ - IF same coplings expect: Br(fi! μ νμ ν fi ) Br(fi! e ν fi ) = 0:9726 gw (the small ifference is e to the slight rection in phase space e to the non-negligible mon mass) The observe ratio 0:974 ± 0:005 is consistent with the preiction Same WEAK CHARGED CURRENT copling for e, μ an fi! LEPTON UNIVERSALITY (see Qestion 9 on the problem sheet)

21 W Leptonic Coplings 2 Stanar Moel W Boson Coplings In the Stanar Moel the charge of the WEAK interactions is calle WEAK ISOSPIN. Leptons are represente in Doblets!!! ψ e νe ; ψ μ νμ ; ψ fi νfi W-bosons only cople particles within a oblet. e.g. no W e νμ copling. τ - ν τ τ - ν τ UNIVERSAL COUPLING STRENGTH g w p 2

22 Weak Interactions of Qarks 22 In the Stanar Moel, the leptonic weak coplings take place within generation, τ - ν τ Natral to expect same Pattern for QUARKS i.e. s t c b Unfortnately its not that simple! Example The ecay K + (μs)! μ + νμ sggests a W + μs copling W + K + s µ +

23 Cabibbo Mixing Angle 23 For-Flavor Qark Mixing the states which take part in the WEAK interaction are ORTHOGONAL combinations of the states of efinite flavor (,s) For 4-flavors, f ; ; s an c g, the mixing can be escribe by a single parameter: the CABIBBO ANGLE c 0 s = 0 Coplings become: cos c sin c sin c cos c s = W- s cosθ c + sinθ c s cosθ c sinθ c s s = W- c cosθ c s - sinθ c cosθ c c - sinθ c c EXPERIMENTALLY: c = 3 ffi

24 EXAMPLE: Nclear Beta Decay 24 Recall: n p Gμ =(:6632 ± 0:00002) 0 5 GeV 2 Gfi =(:36 ± 0:003) 0 5 GeV 2 Strength of copling / gwcos c (Gfi) 2 / jmj 2 / cos 2 c Hence expect Gfi = cos cgμ It works, :6632 cos 3 ffi = :36 Cabibbo Favore : jmj 2 / cos 2 c s c cosθ c cosθ c Cabibbo Sppresse : jmj 2 / sin 2 c s c sinθ c - sinθ c

25 25 EXAMPLE: K +! μ + νμ W + K + s µ + μs copling ) Cabibbo sppresse jmj 2 / sin 2 c EXAMPLE: D 0! K ß +, D 0! K + ß cosθ c cosθ c c π + s D 0 K - sinθ c sinθ c c K + s D 0 π - Expect (D 0! K + ß ) (D 0! K ß + = sin4 c ) cos 4 c ß 0:0028 Measre 0:0038 ± 0:0008 D 0! K + ß is DOUBLY Cabibbo sppresse (see Qestion 8 on the problem sheet)

26 CKM Matrix 26 Cabibbo-Kobayashi-Maskawa Matrix Exten to 3 generations ψ 0 s 0 b 0! = Giving coplings V g w ψ s c V Vs Vb Vc Vcs Vcb Vt Vts Vtb V s g w s t!ψ V b g w s b b! b Note Vckm ß sometimes written 0 Vckm ψ cos c sin c 0:0 sin c cos c 0:05 0:0 0: A! with = sin c (see Qestion 0 on the problem sheet)

27 Lepton Mixing Matrix? 27 Natral to ask if there is an eqivalent of the CKM Matrix for leptons. HYPOTHETICAL EXAMPLE: cosθ l sinθ l The netrinos are nobserve, (i.e. on t istingish the ifferent netrino final states). Conseqently the amplite for μ! e νν jmj 2 / g 2 w (cos2 l + sin 2 l) In the qark sector, mass ifferences between qarks (an the harons they form) allow s to istingish the ifferent final states See Hanot VIII for the evience that there is MIXING in the lepton sector

28 28 Smmary WEAK INTERACTION (CHARGED-CURRENT) Parity violate e to the HELICITY strctre of the interaction Force meiate by massive W-bosons, MW = 80:4 GeV Intrinsically stronger than EM interaction Universal copling to qarks an leptons Qarks take part in the interaction as mixtres of the flavor eigenstates GF=(:6632±0:00002) 0 5 GeV 2 from mon ecay ELECTROWEAK UNIFICATION - next hanot Netral Crrent WEAK interaction - Z 0 Unification of WEAK an EM forces

29 APPENDIX: VECTOR-AXIAL VECTOR (V A) 29 NON-EXAMINABLE In the DIRAC eqation the WEAK interaction vertex has the form VECTOR AXIAL-VECTOR fl μ ( fl 5 ) Consier Dirac spinors for a particle traveling along the z-axis 0 0 R = 0 p (E+m) 0 A L = N 0 0 p (E+m) The WEAK interaction matrix element looks like hμrjfl μ ( fl 5 )jli it has the form VECTOR (fl μ ) mins AXIAL-VECTOR fl μ fl 5 In matrix form: fl 5 = = A A C A A

30 Consier the effect of the interaction on LH an RH spinors: 0 p - (E+m) ( fl 5 B )R = 0 C p -+ A (E+m) 0 ( fl 5 )L = N 0 0 p B+ (E+m) 0 A p -- (E+m) 30 Massless limit, m! 0; p! E: ( fl 5 )R = 0 For massless particles, the form of the interaction projects ot LH particle states, i.e. only LH-particles take part in the WEAK interaction. For massive particles, the form of the interaction preferentially projects ot LH particles. IF netrinos were massless (which is not qite the case), the WEAK coplings of RH netrinos an LH anti-netrinos wol be zero, an if the νr an νl state exist they wol only experience the gravitational interaction!

31 EXAMPLE: ß ± Decay π - Charge Pion ecay branching fractions: ß! μ νμ 99:9877 % ß! e νe 0:023 % 3 Naïvely might expect slightly larger branching fraction for ß! e νe e to phase space! Consier Spin/Helicity π - J P = 0 - Conservation of anglar momentm ) mon an netrino spins in opposite irections. SAME HELICITY Netrinos massless, ) only RH anti-netrino takes part in WEAK interaction therefore μ is also right-hane IF massless, e.g. mμ = 0, the WEAK Matrix element wol be exactly zero

32 ( fl 5 ) R = N 0 p - (E+m) -+ Wrong-Hane ME (zero for m = 0) M / f wrong = 2 0 p (E+m) 0 p E+m C A 32 ß! plept E lept f wrong μ νμ 30 MeV 0 MeV 0.43 e 70 MeV 70 MeV ß! μ νμ is non-relativistic Decay ß! e sppresse relative to ß! μ νμ : ( 0:0035 0:43 )2 ß Once phase-space taken into accont: (ß! e ) (ß! μ = : νμ ) Positron ecay spectrm from ß + ecays. Large Peak ß + μ +! μ + ν! μ e + ν μ Small Peak ß +! e +